You are given the following information about a commercial auto liability book of business:

1. Each insured’s claim count has a Poisson distribution with mean [math]\lambda [/math] , where [math]\lambda [/math] has a gamma distribution with [math]\alpha = 1.5[/math] and [math]\theta = 0.2 [/math].
2. Individual claim size amounts are independent and exponentially distributed with mean 5000.
3. The full credibility standard is for aggregate losses to be within 5% of the expected with probability 0.90.

Calculate the expected number of claims required for full credibility using limited fluctuated credibility.

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Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: B

[math]X[/math] is the random sum [math]Y_{1}+Y_{2}+\cdots+Y_{N}[/math].

[math]N[/math] has a negative binomial distribution with [math]r=a=1.5[/math] and [math]\beta=\theta=0.2[/math].

[[math]]\begin{aligned} & \operatorname{E}(N)=r \beta=0.3, \operatorname{Var}(N)=r \beta(1+\beta)=0.36 \\ & \operatorname{E}(Y)=5,000, \operatorname{Var}(Y)=25,000,000 \\ & \operatorname{E}(X)=0.3(5,000)=1,500 \\ & \operatorname{Var}(X)=0.3(25,000,000)+0.36(25,000,000)=16,500,000 \end{aligned}[[/math]]

Number of exposures (insureds) required for full credibility

[[math]]n_{F U L L}=(1.645 / 0.05)^{2}\left(16,500,000 / 1,500^{2}\right)=7,937.67 \text {. }[[/math]]

Number of expected claims required for full credibility

[[math]]\operatorname{E}(N) n_{F U L L}=0.3(7,937.67)=2,381[[/math]]

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Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.